# Adaptive Selection of Primal Constraints for Isogeometric BDDC Deluxe Preconditioners

- 2017

#### Abstract

Isogeometric analysis has been introduced as an alternative to finite element methods in order to simplify the integration of CAD software and the discretization of variational problems of continuum mechanics. In contrast with the finite element case, the basis functions of isogeometric analysis are often not nodal. As a consequence, there are fat interfaces which can easily lead to an increase in the number of interface variables after a decomposition of the parameter space into subdomains. Building on earlier work on the deluxe version of the BDDC family of domain decomposition algorithms, several adaptive algorithms are here developed for scalar elliptic problems in an effort to decrease the dimension of the global, coarse component of these preconditioners. Numerical experiments provide evidence that this work can be successful, yielding scalable and quasi-optimal adaptive BDDC algorithms for isogeometric discretizations. 1. Introduction. There has recently been a considerable effort in developing adaptive methods for the selection of primal constraints for BDDC algorithms, including its deluxe variant. The primal constraints of a BDDC or FETI–DP algorithm provide the global, coarse part of such a preconditioner and is of crucial importance for obtaining rapid convergence of these preconditioned conjugate gradient methods for the case of many subdomains. When the primal constraints are chosen adaptively, we aim at selecting a primal space, which for a certain dimension of the coarse space, provides the fastest rate of convergence for the iterative method. In the alternative, we can try to develop criteria which will guarantee that the condition number of the iteration stays below a given tolerance. In this paper, we will consider the use of adaptive algorithms to select the primal constraints and the associated BDDC change of basis for elliptic problems and isoge-ometric analysis. While for lower order finite element approximations, one typically starts out with a small primal space, associated with all the subdomain vertex variables , and then adds primal constraints in order to obtain improved iteration counts, a similar strategy for an isogeometric problem will introduce a primal space that can be quite large especially if we have a high polynomial degree and high regularity inside the patches. This depends on the fact that we have fat vertices, potentially with˚Dipartimento

**DOI:**10.1137/15M1054675